Extrinsic definition

Mainly because it is easy to state, we give a definition of a manifold with corners which corresponds to it being a domain in a manifold without boundary. Since this depends on the construction of an `appropriate', but by no means unique, extension across the boundary it is an extrinsic definition (and therefore somewhat objectionable).

Definition A set X with a space of (real-valued) functions defined on it is a manifold with corners if it has the following interrelated properties.

1. There is a smooth (that is infinitely differentiable) manifold without boundary, Y, and an injection with
2. There is a finite collection of functions such that for any and any collection I of indices such that for all the differentials for are independent in
3. 

We usually write